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Polar coordinate system From Wikipedia, the free encyclopedia (Redirected from Polar coordinates) In mathematics, the polar coordinate system is a two- dimensional coordinate system in which points are given by an angle and a distance from a central point known as the pole (equivalent to the origin in the more familiar Cartesian coordinate system). The polar coordinate system is used in many fields, including mathematics, physics, engineering, navigation and robotics. It is especially useful in situations where the relationship between two points is most easily expressed in terms of angles and distance; in the Cartesian coordinate system, such a relationship can only be found through trigonometric formulae. For many types of curves, a polar equation is the simplest means of representation; for some others, it is the only such means. Contents A polar grid with several angles labeled. History See also History of trigonometric functions It is known that the Greeks used the concepts of angle and radius. The astronomer Hipparchus (190-120 BC) tabulated a table of chord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.[1] In On Spirals, Archimedes describes his famous spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system. There are various accounts of who first introduced polar coordinates as part of a formal coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidge's Origin of Polar Coordinates. [2][3] Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts at about the same time. Saint-Vincent wrote about them privately in 1625 and published in 1647, while Cavalieri published in 1635 with a corrected version appearing in 1653. Cavalieri first utilized polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs. In Method of Fluxions (written 1671, published 1736), Sir Isaac Newton was the first to look upon polar coordinates as a method of locating any point in the plane. Newton examined the transformations between polar coordinates and nine other coordinate systems. In Acta eruditorum (1691), Jacob Bernoulli used a system with a point on a line, called the pole and polar axis respectively. Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates. The actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th century Italian writers. The term appeared in English in George Peacock's 1816 translation of Lacroix's Differential and Integral Calculus.[4][5][6] Alexis Clairaut and Leonhard Euler are credited with extending the concept of polar coordinates to three dimensions. Plotting points with polar coordinates As with all two-dimensional coordinate systems, there are two polar coordinates: r (the radial coordinate) and θ (the angular coordinate, polar angle, or azimuth angle, sometimes represented as φ or t). The r coordinate represents the radial distance from the pole, and the θ coordinate represents the anticlockwise (counterclockwise) angle from the 0° ray (sometimes called the polar axis), known as the positive x-axis on the Cartesian coordinate plane.[7] For example, the polar coordinates (3,60°) would be plotted as a point 3 units from the pole on the 60° ray. The coordinates (−3,240°) would also be plotted at this point because a negative radial distance is measured as a positive distance on the opposite ray (240° − 180° = 60°). The points (3,60°) and (4,210°) One important aspect of the polar coordinate system not present in the Cartesian coordinate system is the ability to express a single point with an infinite number of different coordinates. In general, the point (r, θ) can be represented as (r, θ ± n×360°) or (−r, θ ± (2n + 1)180°), where n is any integer.[8] If the r coordinate of a point is 0, then regardless of the θ coordinate, the point will be located at the pole. Use of radian measure Angles in polar notation are generally expressed in either degrees or radians, using the conversion 2π rad = 360°. The choice depends largely on the context. Navigation applications use degree measure, while some physics applications (specifically rotational mechanics) use radian measure, based on the ratio of the radius of the circle to its circumference.[9] Converting between polar and Cartesian coordinates The two polar coordinates r and θ can be converted to Cartesian coordinates by From those two formulas, conversion formulas in terms of x and y are derived, including [10]If x = 0, then if y is positive θ = 90° (π/2 radians) and if y is negative θ = 270° (3π/2 radians). Polar equations The equation of a curve expressed in polar coordinates is known as a polar equation, and is usually written with r as a function of θ. Polar equations may exhibit different forms of symmetry. If r(−θ) = r(θ) then the curve will be symmetrical about the horizontal (0°/180°) ray; if r(π−θ) = r(θ) then it will be symmetric about the vertical (90°/270°); if r(θ−α) = r (θ) then it will be rotationally symmetric α° counterclockwise about the pole.[10] Circle The general equation for any circle with a center at (r0, φ) and radius a is This can be simplified in various ways, to conform to more specific cases, such as the equation for a circle with a center at the pole and radius a.[11] A circle with equation r(θ) = 1. Line Radial lines (those which run through the pole) are represented by the equation , where φ is the angle of elevation of the line; that is φ = arctan m with m the slope of the line in the Cartesian coordinate system. Any line which does not run through the pole is perpendicular to some radial line.[12] The line which crosses the line θ = φ perpendicularly at the point (r0, φ) has equation . Polar Rose A polar rose is a famous mathematical curve which looks like a petalled flower, and which can only be expressed as a polar equation. It is given by the equations OR If k is an integer, these equations will produce a k-petalled rose if k is odd, or a 2k-petalled rose if k is even. If k is not an integer, a disc is formed, as the number of petals is also not an integer. Note that with these equations it is impossible to make a rose with 2 more than a multiple of 4 (2, 6, 10, etc.) petals. The variable a represents the length of the petals of the rose. Archimedean spiral A polar rose with equation r (θ) = 2 sin 4θ. The Archimedean spiral is a famous spiral that was discovered by Archimedes, which also can be expressed only as a polar equation. It is represented by the equation: . Changing the parameter a will turn the spiral, while b controls the distance between the arms, which is always constant.. The Archimedean spiral has two arms, one for θ > 0 and one for θ < 0. The two arms are smoothly connected at the pole. Taking the mirror image of one arm across the 90°/270° line will yield the other arm. One arm of an Archimedean spiral with equation r(θ) = θ for Conic sections 0 < θ < 6π. A Conic section with one focus on the orgin, and the other somewhere on the 0° ray (i.e. the major axis lies along the polar axis) is given by: Ellipse, showing semi-latus rectum where e is the eccentricity and l is the semi-latus rectum, the perpendicular distance at a focus from the major axis to the curve. If e > 1 it defines a hyperbola; if e = 1 it defines a parabola; and if e < 1 this equation defines an ellipse. The special case e = 0 of the latter results in a circle of radius l. Other curves Because of the circular nature of the polar coordinate system, it is much simpler to describe many curves with an equation in polar rather than Cartesian form. Among these curves are lemniscates, limaçons, and cardioids. Complex numbers Complex numbers, written in rectangular form as a + bi, can also be expressed in polar form in two different ways: 1. , abbreviated 2. which are equivalent as per Euler's formula.[13] To convert between rectangular and polar complex numbers, the following conversion formulas are used: and therefore For the operations of multiplication, division, exponentiation, and finding roots of complex numbers, it is much easier to use polar complex numbers than rectangular complex numbers. In abbreviated form: Multiplication: Division: Exponentiation (De Moivre's formula): Vector calculus Calculus can be applied to equations expressed in polar coordinates. Let be the position vector , with r and θ depending on time t, be a unit vector in the direction and be a unit vector at right angles to . The first and second derivatives of position are . Let be the area swept out by a line joining the focus to a point on the curve. In the limit is half the area of the parallelogram formed by and , , and the total area will be the integral of with respect to time. Applications Kepler's laws of planetary motion Further information: Kepler's laws of planetary motion Polar coordinates provide a natural means of expressing Kepler's laws of planetary motion.
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